Kodaira-Spencer maps for elliptic orbispheres as isomorphisms of Frobenius algebras

TL;DR

This paper demonstrates that for elliptic orbispheres, the Kodaira-Spencer map provides an isomorphism between the quantum cohomology and the Jacobian algebra as Frobenius algebras, using Floer theory modifications.

Contribution

It establishes a Frobenius algebra isomorphism via the Kodaira-Spencer map specifically for elliptic orbispheres, advancing the understanding of mirror symmetry in this context.

Findings

Frobenius algebra isomorphism for elliptic orbispheres

Use of Floer theoretic residue pairing modification

Confirmation of mirror symmetry conjecture in this setting

Abstract

Given a mirror pair of a symplectic manifold $X$ and a Landau-Ginzburg potential $W$, we are interested in the problem whether the quantum cohomology of $X$ and the Jacobian algebra of $W$ are isomorphic. Since those can be equipped with Frobenius algebra structures, we might ask whether they are isomorphic as Frobenius algebras. We show that the Kodaira-Spencer map gives a Frobenius algebra isomorphism for elliptic orbispheres, under the Floer theoretic modification of the residue pairing.